The volume of a tetrahedron fomed by the coterminus edges `veca , vecb and vecc is 3` . Then the volume of the parallelepiped formed by the coterminus edges `veca +vecb, vecb+vecc and vecc + veca` is

Options

A. 6
B. 18
C. 36
D. 9

Explanation

To solve the problem step by step, we will use the properties of the volume of a tetrahedron and the volume of a parallelepiped.

### Step 1: Understand the volume of the tetrahedron
The volume \( V \) of a tetrahedron formed by vectors \( \vec{a}, \vec{b}, \vec{c} \) is given by the formula:
\[ V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \]
Given that the volume of the tetrahedron is 3, we can set up the equation:
\[ \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| = 3 \]

### Step 2: Solve for the scalar triple product
Multiplying both sides of the equation by 6 gives:
\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = 18 \]
This means that the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 18 \) (or \( -18 \), but we take the absolute value for volume).

### Step 3: Volume of the parallelepiped
The volume \( V_p \) of the parallelepiped formed by the edges \( \vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a} \) is given by:
\[ V_p = |(\vec{a} + \vec{b}) \cdot ((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}))| \]

### Step 4: Expand the cross product
Using the distributive property of the cross product:
\[ (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}) = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \]
Since \( \vec{c} \times \vec{c} = \vec{0} \), we simplify to:
\[ (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}) = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \]

### Step 5: Calculate the volume of the parallelepiped
Now, substituting back into the volume formula:
\[ V_p = |(\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a})| \]
Using the property of the scalar triple product, we can express this as:
\[ V_p = |(\vec{a} \cdot (\vec{b} \times \vec{c})) + (\vec{b} \cdot (\vec{b} \times \vec{a})) + (\vec{a} \cdot (\vec{c} \times \vec{a}))| \]
The second and third terms vanish because the dot product of a vector with the cross product of itself with another vector is zero. Therefore:
\[ V_p = 2 |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 6: Substitute the known value
Since we found \( |\vec{a} \cdot (\vec{b} \times \vec{c})| = 18 \):
\[ V_p = 2 \times 18 = 36 \]

### Conclusion
Thus, the volume of the parallelepiped formed by the coterminus edges \( \vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a} \) is:
\[ \boxed{36} \]